Question 161327
Let x = length of side of smallest field
then because "the side of one field was 1 kilometer longer than the side of the smallest field"
x+1 = length of one field
and because "the largest field was 3 kilometers longer than the side of the smallest field."
x+3 = length of largest field
.
Area of each field is side^2:
x^2 + (x+1)^2 + (x+3)^2 = 38
expanding through FOIL:
x^2 + x^2+2x+1 + x^2+6x+9 = 38
combining like-terms:
3x^2 + 8x + 10 = 38
3x^2 + 8x - 28 = 0
(3x+14)(x-2) = 0
x = {-14/3, 2}
.
We can toss out the negative solution leaving us with:
x = 2 km
x^2 = 4 sq km (smallest field)
.
(x+1)^2 = x^2+2x+1 = 2^2+2(2)+1 = 4+4+1 = 9 sq km (one field)
.
(x+3)^2 = x^2+6x+9 = 2^2+6(2)+9 = 4+12+9 = 25 sq km (largest field)
.
Answer: 
4 square kilometers
9 square kilometers
25 square kilometers